Topological Subset Space Models for Public Announcements

Abstract

We reformulate a key definition given by Wáng andÅgotnes (2013) to provide semantics for public announcements in subset spaces. More precisely, we interpret the precondition for a public announcement of ϕ to be the "local truth" of ϕ, semantically rendered via an interior operator. This is closely related to the notion of ϕ being "knowable". We argue that these revised semantics improve on the original and offer several motivating examples to this effect. A key insight that emerges is the crucial role of topological structure in this setting. Finally, we provide a simple axiomatization of the resulting logic and prove completeness.

Private Announcements on Topological Spaces

“…Presented by Heinrich Wansing; Received April 4, 2016 be true, but that it be 'observable', as in [12]. It is this observational aspect of our work that makes it different from most approaches to knowledge and semi-private announcements based on standard relational semantics.…”

“…This work follows the tradition of modelling (dynamic) epistemic logics on spatial, rather than relational, structures, such as neighbourhood frames [23,37], subset spaces [1,10,11,17,18,35,36], and more importantly topological spaces [2,3,12,30,31]. Unlike the rather standard approaches to modelling knowledge and information dynamics using relational semantics (see, e.g., [9,26,29,32] for a survey on this topic), our work acknowledges the observation based nature of these notions that demands richer, neighbourhoodlike structures.…”

“…In spite of the intuitive connections between SSL and the informational attitudes studied in DEL, connecting SSL to DEL is not entirely straightforward, even in the relatively simple case of public announcements. Connections between single-agent public announcement logic and SSL were made in [1,7,[10][11][12]36]. To the best of our knowledge, Wáng andÅgotnes [36] were the first to propose semantics for public announcements on subset spaces in terms of open set refinement rather than model restriction.…”

“…To the best of our knowledge, Wáng andÅgotnes [36] were the first to propose semantics for public announcements on subset spaces in terms of open set refinement rather than model restriction. Bjorndahl [12] then proposed a revision of the semantics in [36], based on topological spaces, and with an interior modality int(ϕ) capturing the precondition of an announcement, associated with the interior operator of topological spaces. In previous work [30], we further extended the proposal in [12] with the arbitrary public announcement modality capturing information change caused by any announcement (rather than any effort) and studied a particular type of 'effort' that is in the shape of public announcements.…”

“…Bjorndahl [12] then proposed a revision of the semantics in [36], based on topological spaces, and with an interior modality int(ϕ) capturing the precondition of an announcement, associated with the interior operator of topological spaces. In previous work [30], we further extended the proposal in [12] with the arbitrary public announcement modality capturing information change caused by any announcement (rather than any effort) and studied a particular type of 'effort' that is in the shape of public announcements. Baltag et al [7] further investigated the relationship between the dynamic notions effort, public announcements, and arbitrary announcements, building on the settings presented in [12,30].…”